Saturday X-tra X-Sheet: 8. Inverses and Functions

Transcription

1 Saturda X-tra X-Sheet: 8 Inverses and Functions Ke Concepts In this session we will ocus on summarising what ou need to know about: How to ind an inverse. How to sketch the inverse o a graph. How to restrict the domain o a unction so the inverse is a unction. Terminolog & deinitions From our understanding o smmetr that is undoing what does. Is the relection o about the line =. Smbols, Units & Equations is used to indicate the inverse o. X-planation When inding the equation o we simpl swap the and values around and work the new equation into orm. When sketching we take the coordinates o and interchange the and and then plot these new points to get the graph o the inverse. X-ample Questions. The inverse o a unction is ( ) 2 4. What is the unction ()? (3) 2. The sketch represents the graph o the parabola, which intersects the ais at ( m ; 0) and ( 2 ; 0 ). It is urther given that ; n is the turning point o the parabola. Points ( 0 ; 6 ) 4 and ( k ; 6 ) also lie on the curve o. ; n 4 6 ( k ; 6 ) (m;0) 0 (2;0)

2 (a) the value o k () (b) the value o m () (c) the value o n, showing all working detail. (5) (d) the restriction on the range so that the inverse o is a unction. () 3. The graph o unction g is shown below. (a) Give the domain and range o the unction. (b) Draw a neat sketch graph o the inverse unction o g. (3) (c) Eplain wh this inverse is a unction.

3 4. The graphs o g() = below. 2 and () = 2 2 are drawn on the same set o aes g B E 3; 2 3 A g C (2; -8) (a) (b) Write the equation o Restrict the domain o so that in the orm =. (3) is a unction. () (c) Write the equation o g in the orm =... (d) What do ou notice about g and g ()

4 5. Stud the sketch below, then answer the questions that ollow: Y X (a) Describe the domain o. (b) Sketch the graph o the inverse o, on the set o aes provided. (4) (c) Is a unction or a non-unction? Eplain. (d) For the straight line, when 3;0, write down the equation o orm:, in the.. (e) I the equation o, when 0;4 is: o, write down the equation g which is the relection o about the - ais.

5 6. Use the given sketch o a unction k to answer the ollowing questions: Y X ) Wh does k represent a unction? () 2) Is k one-to-one or a man-to-one unction? Eplain. 3) What is the range o k? 4) Will k represent a unction? Eplain. 7. Sketch the graph o ( ) 3. (a) Label at least two deining points clearl. (b) Determine the inverse o and write it in the orm ( )... () (c) Sketch the graph o on the same set o aes as. Label at least two deining points clearl. (d) Is a unction? Give a reason or our answer. (e) I g() = ( - 4) 3 (i) Describe the transormation o to g. () (ii) Sketch the graph o g. Show an new asmptotes and label at least two deining points clearl. (3)

6 8. The graph is represented b = log a. Q 0 P R X-ercise. 2 (3) (b) I a = 4, determine the value o b i R is the point ( b ; -2 ) (3) (c) The inverse o the graph above is translated up b two units and to the let b one unit. Give the equation o the new graph in the orm = (3) The graphs o : = log 2 and g : 2 + = 2 are shown on the sketch. A and C are points on g and M and B are points on. A is on the -ais and ABCD is a rectangle. Determine: a) the equation o g, the inverse o g, in the orm = b) the co-ordinates o A and hence the co-ordinates o B. (3) c) the co-ordinates o M. (3)

7 2. The sketch represents and a) is a non-unction. Eplain this. () b) Determine a wa in which the domain o should be restricted so that will be a unction. 3. I () = ¼ ², give the equation o in the orm = 4. The ollowing unctions are given : ( ) log 3 and g ( ) (a) Sketch () and g() on the same set o aes. Label all the intercepts with the aes as well as the asmptotes where applicable. (5) (b) Determine the equation o the inverse o () in the orm - () =. (c) Determine the equation o the inverse o g() in the orm g - () = (3) 5. Below is a sketch o. A is a turning point with coordinates ( 5;6). B is the point where the parabolic curve and the straight line meet. The coordinates o B are ( ;2). A B (a) Is a unction? Eplain. (b) Give the values o or which is strictl increasing. (c) Determine the range o. (d) Give one possible restriction on the domain that would ensure that is a unction. (e) Determine (3). ()

8 Answers a) b) c) 2a) 2b) 2 A (0;2) B (4;2) 64 Yes it is a unction the values are not repeated Either 0 or a) See attached sheet 4b) 4c) g ( ) 3 ( ) 5a) Yes it is a unction the values do not repeat. 5b) ( ; 5) 5c) ( ;6) 5d) 5 or 5 5e) (3) 2